Random coding for wireless multicast

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Random coding for wireless multicast
Performance for fading channels and inter-session
coding

• Brooke Shrader and Anthony Ephremides
• University of Maryland
• Joint work with Randy Cogill, University of
  Virginia
• May 9, 2008

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Introduction and Motivation

• Alternatives to overcome channel MAC-layer errors
  in multicast transmission
• repeatedly send packets (ARQ)
• network coding

• Previous work
• network coding outperforms ARQ for time-invariant
  channels
• coding used within (but not between) multicast
  sessions

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Random linear coding for multicast
Form random linear combinations of K packets.
Transmit coefficients ai in packet header Decode
solve a system of linear equations in si
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Multicast throughput
Let Tm denote the number of slots needed for
destination m to collect K linearly independent
random linear combinations. The multicast
throughput is
Difficulty Tm are correlated due to correlation
in the random linear combinations sent to
different destination nodes. This is true even
if the channels to the destination nodes are
independent.
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Lower bound on multicast throughput
Assume the channels to the M destinations are
identically distributed (but not independent).
For random variables X1,X2,,XM identically
distributed and correlated and for any t gt 0,
Then the multicast throughput is lower bounded,
for any t gt 0, as
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Our contributions

• We use this bound to quantify the multicast
  throughput for random linear coding
• over a fading wireless channel where reception
  probability depends on packet length, overhead,
  SNR
• across multiple multicast sessions

• Random linear network coding naturally adapts
  coding rate to variations in the channel.
• Coding across sessions means that receivers
  decode additional packets that arent intended
  for them.

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I Packet length and overhead

• Network coding can approach min-cut capacity in
  the limit as alphabet size approaches infinity.
• Random network coding overhead needed to
  transmit coefficients of random code.
• Packet length (symbols per packet alphabet
  size) must be sufficiently large in order to
• approach min-cut capacity
• ensure small (fractional) overhead

Our approach model the packet erasure
probability as a function of packet length
(symbols per packet and alphabet size).
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I Packet erasure probability
Assume that there is no channel coding within
packets. For packet to be received, every symbol
must be received.
q Probability that a transmitted packet is
successfully received at a destination node. Pu
u-ary symbol error probability for modulation
scheme, depends on SNR, channel model (e.g.,
AWGN)
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I Accounting for overhead
Each packet consists of n u-ary symbols. Coding
is performed on groups of K packets.

The multicast throughput is lower bounded, for
any t gt 0, as
ratio of information to informationoverhead
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I Fading channel model
The channel to each destination node evolves as a
Markov chain with Good and Bad states.
qG Probability that a transmitted packet is
received in Good state qB Probability that a
transmitted packet is received in Bad state A
packet-erasure version of the Gilbert channel
model.
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I Augmented Markov chain for reception at each
destination
State (S,j) where S is Good or Bad state and
j0,1,K is the number of linearly independent
random linear combinations that have been
received.
P transition probability matrix Assume qB0, so
initial state is always SG. Transmission time
T1 time to reach state (S,K) from (G,0).
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I Multicast throughput versus K
Compare to time-invariant channel with
probability of reception
M10, n250, u8 QAM modulation over AWGN channel
with SNR/bit 3.5 dB in Good state and -8 dB in
Bad state.
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II Coding across multicast sessions

• One source node
• K multicast sessions, each with an independent
  arrival process of equal rate
• Each session serves M destination nodes
• Channels to all MK destination nodes are
  identically distributed with reception
  probability q

Random linear coding create random linear
combinations from the K head-of-line packets, one
from each session.
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II Coding across multicast sessions

• For successful decoding, each destination must
  decode the packets from all K multicast sessions.
• Using bound on Emax(X1,X2,XMK), we bound the
  throughput as

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II Multicast throughput for coding across
sessions
For large number of sessions and receivers per
session, coding outperforms retransmissions
K50, q0.8
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Conclusions

• We provided a lower bound on multicast throughput
  for random linear coding while accounting for
  packet length, overhead, SNR, and fading.
• We demonstrated that random linear coding across
  multiple multicast sessions often outperforms
  ARQ.
• Future work
• incorporate channel coding within packet, study
  how to allocate coding within and among packets
• code over multiple packets from multiple flows